It is already April, but you might still be interested. Some great tips have already been mentioned, but I might add one more.
I am running a Athlon 64 3200+ myself, with 1.5 Gb of ram. I guess it originally had a GeForce FX5200 128 Mb, not 100% sure about that, but I swapped that one for an AMD/ATI HD4650 1GB a long time ago.
I am able to run 720p YouTube clips smoothly.
I can even run 1080p YouTube clips smoothly, but they have to be fully downloaded before playback.
As far as I know Adobe stopped making new versions of Flash for Linux in February 2012. So basically you will get the latest version Adobe provided (11.2), which wasn't great. Please note: Adobe still provides security updates.
Luckily the Flash player that is integrated in the Chrome browser will still be at the latest version and that is currently version 11.6. (Please note: not for the Chromium browser, only Chrome.)
I get the best flash performance with Chrome these days. So you'll might give it a try.
Too bad it limits your choice, but well..
When I upgraded my video-card the video performance went up by a large step, but at this stage I would not recommend buying a new video-card because it would not be wise investing in a PC that old.
Oh, and make sure you run the 64-bit version of Mint (or other distro of choice) to get the most out of your hardware.